1. PBS Kids - Lets kids watch their favorite PBS shows, including Elmo!! I love it because all the PBS shows are entertaining AND educational. For more PBS Kids apps, click here.
2. All of the Fisher Price games and apptivity apps - They're super easy for a toddler to play on his or her own, and they can't download or buy anything from the app.
3. Netflix - You might be careful with this one, because they can access all of your Netflix content, but if you help them find a show, there are tons and tons of kids' videos on there, from full-length Disney animated films to Veggie Tales to Sesame Street. Also, this app is available for almost ANY smartphone/tablet.
4. The Night Sky - My kid has just recently become fascinated by space. She loves to go out and look at the stars or find the moon. This app finds constellations, planets, and satellites just by aiming your device. It's probably best used with adult supervision, but super-fun for all!
5. Not really an app, but you can also load your kids favorite music, pictures, and home videos on to a smartphone. We live far from family, so I like to have pics and videos of family for my daughter to look at so that she can stay to connected to everyone. If you're creative, there's always something new on a handheld device to occupy a toddler or young child.
When used sparingly, having a smartphone/tablet accessible for your child(ren) can be a life-saver. All you need is a "life-proof" case and a few good apps and/or media, and voila! there's your entertainment on-the-go! Make sure you don't allow your little one to use an app that allows downloads or purchases. You don't want any surprise charges on the credit card! And, if you're like me and don't want the little ones messing with your own smart device, make sure you put a lock code on that thing! It frustrates my little one that she can't get into my phone now, but my apps are safe from her little hands.